A079510 → basis A008277

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel

Recurrence

\(P_{n}(x) = \left(1 + x + n + n x\right)\,P_{n-2}(x) + \left(x + n x\right)\,P_{n-2}'(x) + \left(1 + x + n + n x\right)\,P_{n-1}(x) + \left(x + n x\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 2 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	2														4
2	9	21	6													36
3	44	228	164	24												460
4	265	2755	3795	1330	120											8265
5	1854	37650	89160	56250	11724	720										197358
6	14833	580503	2234890	2262925	817600	113148	5040									6028939
7	133496	10023944	60722424	92806000	51726360	12100256	1193184	40320								228745984
8	1334961	192215007	1795287501	3989097126	3220098525	1132419456	185408748	13687056	362880							10529911260
9	14684570	4059463590	57702687260	181836224290	204410275510	101389431570	24507332360	2964209400	169932960	3628800						577057870310
10	176214841	93713143975	2010201631680	8830791864675	13463749107501	9050376282101	3029081426602	533675040316	49616149176	2272422240	39916800					37063693199907
11	2290792932	2348833374492	75617826772620	457421000097600	928241100048240	822945043980444	366322644316860	88160433080280	11816567867280	870559110432	32581630080	479001600				2753779360072860
12	32071101049	63538280581191	3059246212507771	25254257551294200	67272088490632740	77137805819503143	44426947506068127	14025759679206990	2541366935493100	267728491461288	16011691128576	498804359040	6227020800			234065287760358015
13	481066515734	1845320257274858	132586709245155984	1483742095427866274	5133986571067153800	7503507686802155028	5481190986826498944	2207864834681564058	519296002995429660	73348436127086120	6231204712837104	308499423208704	8123284834560	87178291200		22543917039095872028
14	7697064251745	57273453199933215	6132700873662709710	92574470510614584765	412729344761521397610	760276974229316033490	693899565494244415020	349511140338281031825	103776403856086138320	18869985241745952600	2135033175932548320	149327927021679120	6220228225311360	140250118982400	1307674368000	2440118589344709337500

Technical details

Node id: n-dc4cb0efc0b86256 · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-042d8261c9343ce3

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:        1;
  1:        2,          2;
  2:        9,         21,           6;
  3:       44,        228,         164,           24;
  4:      265,       2755,        3795,         1330,          120;
  5:     1854,      37650,       89160,        56250,        11724,          720;
  6:    14833,     580503,     2234890,      2262925,       817600,       113148,        5040;
  7:   133496,   10023944,    60722424,     92806000,     51726360,     12100256,     1193184,      40320;
  8:  1334961,  192215007,  1795287501,   3989097126,   3220098525,   1132419456,   185408748,   13687056,    362880;
  9: 14684570, 4059463590, 57702687260, 181836224290, 204410275510, 101389431570, 24507332360, 2964209400, 169932960, 3628800;

%F Formula

The row polynomials satisfy P(n,x) = (n+1+(n+1)*x)*P(n-2,x) + (n+1)*x*d/dx P(n-2,x) + (n+1+(n+1)*x)*P(n-1,x) + (n+1)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[(n + 1 + (n + 1)*x)*P[n - 2, x] + (n + 1)*x*D[P[n - 2, x], x] + (n + 1 + (n + 1)*x)*P[n - 1, x] + (n + 1)*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (n + 1 + (n + 1)*x)*v[i - 2] + (n + 1)*x*deriv(v[i - 2], x) + (n + 1 + (n + 1)*x)*v[i - 1] + (n + 1)*x*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Ray basis map theorem
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Ray basis map theorem
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Ray basis map theorem

Proof chains ? JSON

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›
- from Triangle T(n,k) read by rows; related to number of preorders.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741